Chapter 25

The value of \(\cos\) a \(\cos 2 a \cos 3 a \ldots . \cos 999 a\), where \(a=\frac{2 \pi}{1999}\), is (A) \(\frac{1}{2^{99}}\) (B) \(\frac{1}{2^{999}}\) (C) \(\frac{1}{2^{9999}}\) (D) \(\frac{1}{2^{1999}}\)

Let \(a_{1}=\left(\tan \frac{\pi}{8}\right)^{\tan \frac{\pi}{8}}, a_{2}=\left(\tan \frac{\pi}{8}\right)^{\cos \frac{\pi}{8}}\), \(a_{3}=\left(\cot \frac{\pi}{8}\right)^{\tan \frac{\pi}{8}}, a_{4}=\left(\cot \frac{\pi}{8}\right)^{\cos \frac{\pi}{8}}\) Then, (A) \(a_{4}>a_{3}>a_{2}>a_{1}\) (B) \(a_{3}>a_{4}>a_{2}>a_{1}\) (C) \(a_{4}>a_{3}>a_{1}>a_{2}\) (D) \(a_{3}>a_{1}>a_{2}>a_{4}\)

If \(x \cos ^{2} 3 \theta+y \cos ^{4} \theta=16 \cos ^{6} \theta+9 \cos ^{2} \theta\) be an iden- tity, then (A) \(x=-1, y=24\) (B) \(x=1, y=24\) (C) \(x=24, y=1\) (D) none of these

\(|\tan \theta+\sec \theta|=|\tan \theta|+|\sec \theta|, 0 \leq \theta \leq 2 \pi\) is possible only if (A) \(\theta \in[0, \pi]-\left\\{\frac{\pi}{2}\right\\}\) (B) \(\theta \in[0, \pi]\) (C) \(\theta \in\left[0, \frac{\pi}{2}\right)\) (D) none of these

If \(\sin \theta, \sin \phi\) and \(\cos \theta\) are in G.P., then the roots of the equation \(x^{2}+2 x \cot \phi+1=0\) are always (A) real (B) imaginary (C) equal (D) greater than 1

If \(\cos 25^{\circ}+\sin 25^{\circ}=k\), then \(\cos 50^{\circ}\) is equal to (A) \(k \sqrt{2-k^{2}}\) (B) \(-\sqrt{2-k^{2}}\) (C) \(\sqrt{2-k^{2}}\) (D) \(-k \sqrt{2-k^{2}}\)

If \(\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}=x\) then \(\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha}=\ldots\) (A) \(\frac{1}{x}\) (B) \(x\) (C) \(1-x\) (D) \(1+x\)

If \(e^{-\pi 2}<\theta<\pi / 2\), then (A) \(\cos \log \theta<\log \cos \theta\) (B) \(\cos \log \theta>\log \cos \theta\) (C) \(\cos \log \theta \leq \log \cos \theta\) (D) none of these

If \(x y+y z+z x=1\), then \(\sum \frac{x+y}{1-x y}=\) (A) \(\frac{4}{x y z}\) (B) \(\frac{1}{x y z}\) (C) \(x y z\) (D) none of these

\(\cos 12^{\circ} \cos 24^{\circ} \cos 36^{\circ} \cos 48^{\circ} \cos 72^{\circ} \cos 96^{\circ}\) equals (A) \(-\frac{1}{2^{6}}\) (B) \(\frac{1}{2^{8}}\) (C) \(\frac{1}{2^{7}}\) (D) \(-\frac{1}{2^{7}}\)

If \(\alpha, \beta, \gamma \in\left(0, \frac{\pi}{2}\right)\), then \(\frac{\sin (\alpha+\beta+\gamma)}{\sin \alpha+\sin \beta+\sin \gamma}\) is \((\mathrm{A})<1\) (B) \(>1\) \((\mathrm{C})=1\) (D) none of these

If \(\left|\cos \theta\left\\{\sin \theta+\sqrt{\sin ^{2} \theta+\sin ^{2} \alpha}\right\\}\right| \leq k\), then the value of \(k\) is (A) \(\sqrt{1+\cos ^{2} \alpha}\) (B) \(\sqrt{1+\sin ^{2} \alpha}\) (C) \(\sqrt{2+\sin ^{2} \alpha}\) (D) \(\sqrt{2+\cos ^{2} \alpha}\)

The maximum value of \(\left(\cos \alpha_{1}\right)\left(\cos \alpha_{2}\right) \cdot \dot{\pi} .(\cos a n)\) under the restrictions \(0 \leq \alpha_{1}, \alpha_{2}, \ldots, \alpha_{n} \leq \frac{\pi}{2}\) and (cot \(\left.\alpha_{1}\right)\left(\cot \alpha_{2}\right) \ldots\left(\cot \alpha_{n}\right)=1\) is (A) \(\frac{1}{2^{n / 2}}\) (B) \(\frac{1}{2^{n}}\) (C) \(\frac{1}{2 n}\) (D) 1

The inequality \(2^{\sin \theta}+2^{\cos \theta} \geq 2^{\left(t-\frac{1}{\sqrt{2}}\right)}\) holds for (A) \(0 \leq \theta<\pi\) (B) \(\pi \leq \theta<2 \pi\) (C) for all real \(\theta\) (D) none of these

The expression \(2^{\sin \theta}+2^{-\cos \theta}\) is minimum when \(\theta\) is equal to (A) \(2 n \pi+\frac{\pi}{4}, n \in I\) (B) \(2 n \pi+\frac{7 \pi}{4}, n \in I\) (C) \(n \pi \pm \frac{\pi}{4}, n \in I\) (D) none of these

If \(x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}\), then \(\cos \left(\frac{\sqrt{1-x_{0}^{2}}}{x_{1} x_{2} x_{3} \ldots . \text { to } \infty}\right)\left(-1

If \(0<\theta<\pi\), then (A) \(1+\cot \theta \leq \cot \frac{\theta}{2}\) (B) \(1+\cot \theta \geq \cot \frac{\theta}{2}\) (C) \(1+\cot \frac{\theta}{2} \geq \cot \theta\) (D) \(1+\cot \frac{\theta}{2} \leq \cot \theta\)

If \(\cos (\theta-\alpha)=a\) and \(\sin (\theta-\beta)=b(0<\theta-\alpha, \theta-\beta<\) \(\pi / 2\) ), then \(\cos ^{2}(\alpha-\beta)+2 a b \sin (\alpha-\beta)\) is equal to (A) \(a^{2}-b^{2}\) (B) \(a^{2}+b^{2}\) (C) \(2 a^{2} b^{2}\) (D) \(a^{2} b^{2}\)

If in the triangle \(A B C, \tan \frac{A}{2}, \tan \frac{B}{2}\) and \(\tan \frac{C}{2}\) are in harmonic progression, then the least value of \(\cot \frac{B}{2}\) is (A) \(\sqrt{2}\) (B) \(\sqrt{3}\) (C) 2 (D) none of these

If \(x \sin a+y \sin 2 a+z \sin 3 a=\sin 4 a x \sin b+\) \(y \sin 2 b+z \sin 3 b=\sin 4 b x \sin c+y \sin 2 c+z\) \(\sin 3 c=\sin 4 c\), then the roots of the equation \(t^{3}-\frac{z}{2} t^{2}-\frac{y+2}{4}+\frac{z-x}{8}=0 ; a, b, c \neq n \pi\), are (A) \(\sin a, \sin b, \sin c\) (B) \(\cos a, \cos b, \cos c\) (C) \(\sin 2 a, \sin 2 b, \sin 2 c\) (D) \(\cos 2 a, \cos 2 b, \cos 2 c\)

If \(a \sin x+b \cos (x+\theta)+b \cos (x-\theta)=d\), then the minimum value of \(|\cos \theta|\) is (A) \(\frac{1}{2|a|} \sqrt{d^{2}-a^{2}}\) (B) \(\frac{1}{2|b|} \sqrt{d^{2}-a^{2}}\) (C) \(\frac{1}{2|b|} \sqrt{a^{2}-d^{2}}\) (D) none of these

If \(\sin \theta+\cos \theta=\frac{\sqrt{7}}{2}\) and \(0<\theta<\pi / 6\), then \(\tan \left(\frac{\theta}{2}\right)\) equals (A) \(\sqrt{7}-2\) (B) \(\frac{1}{3}(\sqrt{7}-2)\) (C) \(2-\sqrt{7}\) (D) \(\frac{1}{3}(2-\sqrt{7})\)

If \(\sin (\theta+\alpha)=a\) and \(\sin \left(\theta+\beta=b\left(0<\alpha, \beta, \theta<\frac{\pi}{2}\right)\right.\) then \(\cos ^{2}(\alpha-\beta)-4 a b \cos (\alpha-\beta)\) is equal to (A) \(1-2 a^{2}-2 b^{2}\) (B) \(1+2 a^{2}+2 b^{2}\) (C) \(1-a^{2}-b^{2}\) (D) none of these

If \(\sin x+\operatorname{cosec} x+\tan y+\cot y=4\), where \(x\) and \(y \in\left[0, \frac{\pi}{2}\right]\), then \(\tan \frac{y}{2}\) is a root of the equation (A) \(\alpha^{2}+2 \alpha+1=0\) (B) \(\alpha^{2}+2 \alpha-1=0\) (C) \(2 \alpha^{2}-2 \alpha-1=0\) (D) none of thes

The value of \(\cos \theta \cdot \cos 2 \theta \cdot \cos 2^{2} \theta \ldots \cos 2^{n-1} \theta\) for \(\theta=\frac{\pi}{2^{n}+1}\) is (A) 1 (B) \(\frac{1}{2^{n}}\) (C) \(2^{n}\) (D) none of these

The sum of the series \(\sin \theta \cdot \sec 3 \theta+\sin 3 \theta \cdot \sec 3^{2} \theta+\) \(\sin 3^{2} \theta \sec 3^{3} \theta+\ldots\) up to \(n\) terms is (A) \(\frac{1}{2}\left(\tan 3^{n} \theta-\tan \theta\right)\) (B) \(\left(\tan 3^{n} \theta-\tan \theta\right)\) (C) \(\tan 3^{n} \theta-\tan 3^{n-1} \theta\) (D) none of these

If \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) are in A.P. whose common difference is \(\alpha\), then the value of \(\sin \alpha\left[\sec x_{1} \sec x_{2}+\sec x_{2} \sec \right.\) \(\left.x_{3}+\cdots+\sec x_{n-1} \sec x_{n}\right]\) is equal to (A) \(\frac{\sin n \alpha}{\cos x_{1} \cos x_{n}}\) (B) \(\frac{\sin (n-1) \alpha}{\cos x_{1} \cos x_{n}}\) (C) \(\frac{\sin (n+1) \alpha}{\cos x_{1} \cos x_{n}}\) (D) none of these

If \(\frac{\sin \alpha}{\sin \beta}=\frac{\sqrt{3}}{2}\) and \(\frac{\cos \alpha}{\cos \beta}=\frac{\sqrt{5}}{2}, 0<\alpha<\beta<\frac{\pi}{2}\), then (A) \(\tan \alpha=1\) (B) \(\tan \alpha=\frac{\sqrt{3}}{\sqrt{5}}\) (C) \(\tan \beta=\frac{\sqrt{3}}{\sqrt{5}}\) (D) \(\tan \beta=1\)

If \(\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec} x=7\) and \(\sin 2 x=a-b \sqrt{7}\), then (A) \(a=8\) (B) \(b=22\) (C) \(a=22\) (D) \(b=8\)

Let \(n\) be an odd integer. If \(\sin n \theta=\sum_{r=0}^{n} b_{r} \sin ^{r} \theta\), for every value of \(\theta\), then (A) \(b_{0}=0\) (B) \(b_{0}=n\) (C) \(b_{1}=0\) (D) \(b_{1}=n\)

Let \(n\) be a fixed positive integer such that \(\sin \left(\frac{\pi}{2 n}\right)+\cos \left(\frac{\pi}{2 n}\right)=\frac{\sqrt{n}}{2}\), then (A) \(n=4\) (B) \(n=5\) (C) \(n=6\) (D) none of these

If \(a \cos ^{2} 3 \alpha+b \cos ^{4} \alpha=16 \cos ^{6} \alpha+9 \cos ^{2} \alpha\) is iden- tity, then (A) \(a=1\) (B) \(a=24\) (C) \(b=1\) (D) \(b=24\)

If \(A\) and \(B\) are acute angle such that \(A+B\) and \(A-B\) satisfy the equation \(\tan ^{2} \theta-4 \tan \theta+1=0\), then (A) \(A=\frac{\pi}{4}\) (B) \(B=\frac{\pi}{6}\) (C) \(A=\frac{\pi}{6}\) (D) \(B=\frac{\pi}{4}\)

For \(0<\phi<\pi / 2\), if \(x=\sum_{n=0}^{\infty} \cos ^{2 n} \phi, y=\sum_{n=0}^{\infty} \sin ^{2 n} \phi\), and \(z=\sum_{n=0}^{x} \cos ^{2 n} \phi \sin ^{2 n} \phi\), then \(x y z=\) (A) \(x y+z\) (B) \(x z+y\) (C) \(x+y+z\) (D) \(y z+x\)

Let \(f_{n}(\theta)=\tan \frac{\theta}{2}(1+\sec \theta)(1+\sec 2 \theta)(1+\sec 4 \theta) \ldots .\) \(\left(1+\sec 2^{n} \theta\right)\), then (A) \(f_{2}\left(\frac{\pi}{16}\right)=1\) (B) \(f_{3}\left(\frac{\pi}{32}\right)=1\) (C) \(f_{4}\left(\frac{\pi}{64}\right)=1\) (D) \(f_{5}\left(\frac{\pi}{128}\right)=1\)

If \((a-b) \sin (\theta+\phi)=(a+b) \sin (\theta-\phi)\) and \(a \tan \frac{\theta}{2}-b \tan \frac{\phi}{2}=c\), then (A) \(b \tan \phi=a \tan \theta\) (B) \(a \tan \phi=b \tan \theta\) (C) \(\sin \phi=\frac{2 b c}{a^{2}-b^{2}-c^{2}}\) (D) \(\sin \theta=\frac{2 a c}{a^{2}-b^{2}+c^{2}}\)

If \(\alpha, \beta\) and \(\gamma\) are connected by the relation \(2 \tan ^{2} \alpha\) \(\tan ^{2} \beta \tan ^{2} \gamma+\tan ^{2} \alpha \tan ^{2} \beta+\tan ^{2} \beta \tan ^{2} \gamma+\tan ^{2} \gamma \tan ^{2} \alpha=\) 1 , then (A) \(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1\) (B) \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=2\) (C) \(\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=1\) (D) \(\cos (\alpha+\beta) \cos (\alpha-\beta)=-\cos ^{2} \gamma\)

Column-I I. The value of \(\frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}\) is II. If \(A\) and \(B\) be acute positive angles satisfying \(3 \sin ^{2} A+2 \sin ^{2} B=1\) and \(3 \sin\) \(2 A-2 \sin 2 B=0\), then \(\cos (A+2 B)=\) III. The number of integral values of \(k\) for which the equation \(7 \cos x+5 \sin x=\) \(2 k+1\) has a solution is IV. If \(A=\tan 27 \theta-\tan \theta\) and \(B=\frac{\sin \theta}{\cos 3 \theta}+\frac{\sin 3 \theta}{\cos 9 \theta}+\frac{\sin 9 \theta}{\cos 27 \theta}\) then \(\frac{A}{B}=\) Column-II (A) \(\frac{2}{1}\) (B) \(\frac{1}{16}\) (C) 0 (D) 8

\(\sin ^{2} \theta=\frac{4 x y}{(x+y)^{2}}\) is true if and only if: (A) \(x+y \neq 0\) (B) \(x=y, x \neq 0, y \neq 0\) (C) \(x=y\) (D) \(x \neq 0, y \neq 0\)

The value of \(\frac{1-\tan ^{2} 15^{\circ}}{1+\tan ^{2} 15^{\circ}}\) is: (A) 1 (B) \(\sqrt{3}\) (C) \(\frac{\sqrt{3}}{2}\) (D) 2

If \(\sin (\alpha+\beta)=1, \sin (\alpha-\beta)=\frac{1}{2}\), then \(\tan (\alpha+2 \beta)\) tan \((2 \alpha+\beta)\) is equal to: (A) 1 (B) \(-1\) (C) zero (D) none of these

If \(y=\sin ^{2} \theta+\operatorname{cosec}^{2} \theta, \theta \neq 0\) then: (A) \(y=0\) (B) \(y \leq 2\) (C) \(y \geq-2\) (D) \(y \geq 2\)

In a triangle \(A B C, a=4, b=3, \angle A=60^{\circ}\), then \(c\) is the root of the equation: (A) \(c^{2}-3 c-7=0\) (B) \(c^{2}+3 c+7=0\) (C) \(c^{2}-3 c+7=0\) (D) \(c^{2}+3 c-7=0\)

In a \(\triangle A B C, \tan \frac{A}{2}=\frac{5}{6}, \tan \frac{C}{2}=\frac{2}{5}\), then: (A) \(a, c, b\) are in \(\mathrm{AP}\) (B) \(a, b, c\) are in \(\mathrm{AP}\) (C) \(b, a, c\) are in \(\mathrm{AP}\) (D) \(a, b, c\) are in GP

The equation \(a \sin x+b \cos x=c\) where \(|c|>\sqrt{a^{2}+b^{2}}\) has: (A) a unique solution (B) infinite number of solutions (C) no solution (D) none of the above

If \(\alpha\) is a root of \(25 \cos ^{2} \theta+5 \cos \theta-12=0 \frac{\pi}{2}<\alpha<\pi\) then \(\sin 2 \alpha\) is equal to : (A) \(\frac{24}{25}\) (B) \(-\frac{24}{25}\) (C) \(\frac{13}{18}\) (D) \(-\frac{13}{18}\)

If in a triangle \(A B C a \cos ^{2}\left(\frac{C}{2}\right)+c \cos ^{2}\left(\frac{A}{2}\right)=\frac{3 b}{2}\) then the sides \(a, b\) and \(c\) (A) are in A.P. (B) are in G.P. (C) are in H.P. (D) satisfy \(a+b=c\)

Let \(\alpha, \beta\) be such that \(\pi<\alpha-\beta<3 \pi\). If \(\sin \alpha+\sin \beta=-\frac{21}{65}\) and \(\cos \alpha+\cos \beta=-\frac{27}{65}\), then the value of \(\cos \frac{\alpha-\beta}{2}\) is (A) \(-\frac{3}{\sqrt{130}}\) (B) \(\frac{3}{\sqrt{130}}\) (C) \(\frac{6}{65}\) (D) \(-\frac{6}{65}\)

If \(u=\sqrt{a^{2} \cos ^{2} \theta+b^{2} \sin ^{2} \theta}+\sqrt{a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta}\), then the difference between the maximum and minimum values of \(u^{2}\) is given by (A) \(2\left(a^{2}+b^{2}\right)\) (B) \(2 \sqrt{a^{2}+b^{2}}\) (C) \((a+b)^{2}\) (D) \((a-b)^{2}\)

The sides of a triangle are \(\sin \alpha, \cos \alpha\) and \(\sqrt{1+\sin \alpha \cos \alpha}\) for some \(0<\alpha<\frac{\pi}{2}\). Then the greatest angle of the triangle is (A) \(60^{\circ}\) (B) \(90^{\circ}\) (C) \(120^{\circ}\) (D) \(150^{\circ}\)